#8038984

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,4)(5,6)(7,8)(10,13)(14,16)(17,18)(19,20)(21,29)(22,24)(23,27)(25,30)(26,31)(28,32);(1,3,6,7)(2,4,5,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,24,20)(21,26,30,32)(25,31,29,28);(1,2,6,5)(3,8,7,4)(9,11,12,15)(10,14,13,16)(18,24)(20,23)(25,29)(28,31);(1,4)(2,7)(3,5)(6,8)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,32)(30,31);(1,4)(2,3)(5,7)(6,8)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,31)(30,32);(1,5,6,2)(3,8,7,4)(17,21,22,30)(18,25,24,29)(19,26,27,32)(20,28,23,31)

Polycyclic presentation:

2^9|b^2=h,c^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265947, #72265951, #72265957
maximal normal subgroup of	#72265947, #72265951, #72265957
maximal quotient of	#72266510, #72266520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7946180);